1 Introduction
Let
$\mathbb{F}_{q}$
be a finite field with
$q$
elements, where
$q$
is odd, and let
$\mathbb{F}_{q}[x]$
be the polynomial ring over
$\mathbb{F}_{q}$
in the variable
$x$
. We denote by
${\mathcal{H}}_{2g+1,q}$
the set of monic, square-free polynomials
$D\in \mathbb{F}_{q}[x]$
of degree
$2g+1$
. This is the hyperelliptic ensemble of the title. Associated with each
$D$
is a non-trivial quadratic Dirichlet character
$\unicode[STIX]{x1D712}_{D}$
and a quadratic Dirichlet
$L$
-function, which is the same as the Artin
$L$
-function corresponding to the character
$\unicode[STIX]{x1D712}_{D}$
of
$\mathbb{F}_{q}(x)(\sqrt{D(x)})$
, where
$\mathbb{F}_{q}(x)$
is the rational function field over
$\mathbb{F}_{q}$
. These functions will be described more fully in the next section, but, in order to explain the contents of this paper, we introduce some of the basic notation now. Excellent general references are Rosen [Reference Rosen14] and Thakur [Reference Thakur15].
If
$f$
is a non-zero polynomial in
$\mathbb{F}_{q}[x]$
, we define the norm of
$f$
to be
$|f|=q^{\text{deg}\,f}$
. If
$f=0$
, we set
$|f|=0$
. A monic irreducible polynomial
$P$
is called a prime polynomial or simply a prime. The
$L$
-function corresponding to the quadratic character
$\unicode[STIX]{x1D712}_{D}$
is given by the Euler product
$$\begin{eqnarray}L(s,\unicode[STIX]{x1D712}_{D})=\mathop{\prod }_{P\,\text{prime}}(1-\unicode[STIX]{x1D712}_{D}(P)|P|^{-s})^{-1},\quad \operatorname{Re}s>1,\end{eqnarray}$$
where
$s$
is a complex variable. Multiplying out, we obtain the Dirichlet series representation
$$\begin{eqnarray}L(s,\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{f\,\text{monic}}\frac{\unicode[STIX]{x1D712}_{D}(f)}{|f|^{s}},\quad \operatorname{Re}s>1.\end{eqnarray}$$
It is often convenient to work with the equivalent functions written in terms of the variable
$u=q^{-s}$
, namely,
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=\mathop{\prod }_{P\,\text{prime}}(1-\unicode[STIX]{x1D712}_{D}(P)u^{\text{deg}\,P})^{-1},\quad |u|<1/q\end{eqnarray}$$
and
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{f\,\text{monic}}\unicode[STIX]{x1D712}_{D}(f)u^{\text{deg}\,f},\quad |u|<1/q.\end{eqnarray}$$
It turns out that
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
is actually a polynomial of degree
$2g$
(see Rosen [Reference Rosen14, Proposition 4.3]) and it satisfies a Riemann hypothesis (see Weil [Reference Weil17]), namely, all its zeros lie on the circle
$|u|=q^{-1/2}$
. It follows that we may write
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=\mathop{\prod }_{j=1}^{2g}(1-\unicode[STIX]{x1D6FC}_{j}u),\end{eqnarray}$$
where the
$\unicode[STIX]{x1D6FC}_{j}=q^{1/2}e(-\unicode[STIX]{x1D703}_{j}),j=1,2,\ldots ,2g,$
are the reciprocals of the roots
$u_{j}=q^{-1/2}e(\unicode[STIX]{x1D703}_{j})$
of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. (Throughout we write
$e(x)$
to denote
$\text{e}^{2\unicode[STIX]{x1D70B}\text{i}x}$
.) In particular, the restriction
$|u|<1/q$
in (4) (but not in (3)) may be deleted.
Now
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
satisfies the functional equation
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=(qu^{2})^{g}{\mathcal{L}}(1/qu,\unicode[STIX]{x1D712}_{D})\end{eqnarray}$$
and also possesses an “approximate functional equation”
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{\substack{ f\,\text{monic} \\ \text{deg}\,f\leqslant g}}\unicode[STIX]{x1D712}_{D}(f)u^{\deg f}+(qu^{2})^{g}\mathop{\sum }_{\substack{ f\,\text{monic} \\ \text{deg}\,f\leqslant g-1}}\unicode[STIX]{x1D712}_{D}(f)(qu)^{-\text{deg}\,f},\end{eqnarray}$$
which, of course, is exact. The name comes from the analogous formulas in the number field setting which are approximations. For instance, for the Riemann zeta function, a symmetrized version of the formula is (see Titchmarsh [Reference Titchmarsh and Heath-Brown16])
$$\begin{eqnarray}\unicode[STIX]{x1D701}(s)=\mathop{\sum }_{n\leqslant \sqrt{t/2\unicode[STIX]{x1D70B}}}n^{-s}+\unicode[STIX]{x1D712}(s)\mathop{\sum }_{n\leqslant \sqrt{t/2\unicode[STIX]{x1D70B}}}n^{s-1}+E(s),\end{eqnarray}$$
where
$0<\operatorname{Re}\,s<1,\;t\geqslant 1,$
and
$E(s)$
is an error term. The importance of this formula in applications is that it consists of two Dirichlet polynomials of length about
$\sqrt{t}$
, whereas a more direct approximation (see Titchmarsh [Reference Titchmarsh and Heath-Brown16]) would require a Dirichlet polynomial of length
$t$
. The factor
$\unicode[STIX]{x1D712}(s)$
is from the functional equation
$\unicode[STIX]{x1D701}(s)=\unicode[STIX]{x1D712}(s)\unicode[STIX]{x1D701}(1-s)$
and is easy to calculate. Similarly, (7) consists of two pieces of length about
$g$
as opposed to a polynomial of length
$2g$
(recall (5)). This is analogous because, in a sense, large
$t$
in the number field case corresponds to
$q^{2g}$
.
In [Reference Gonek9] and [Reference Gonek10], another type of approximation of the Riemann zeta function and Dirichlet
$L$
-functions was constructed. It was based on the approximate functional equation, but used truncations of the
$L$
-function’s Euler product rather than its Dirichlet series. It was shown, for example, that these approximations satisfy a Riemann hypothesis and are very accurate if one stays away from the zeros of the
$L$
-function. Moreover, if the Riemann hypothesis holds for the
$L$
-function, the zeros of the approximations converge to those of the
$L$
-function as the length of the Euler product tends to infinity. This type of approximation has also been considered in the physics literature; see, for example, [Reference Bogomolny and Keating2].
Our goal in this paper is to carry out a similar construction and analysis in the case of quadratic
$L$
-functions for the hyperelliptic ensemble over finite function fields. An advantage we have in this setting is that the Riemann hypothesis is known to hold for such
$L$
-functions. This means that all our results are unconditional. Moreover, two interesting differences from the zeta and Dirichlet
$L$
-function cases studied in [Reference Gonek9] and [Reference Gonek10] also emerge. The first is that the proof of the hybrid formula in the function field setting is remarkably straightforward, and moreover the result is simple and exact. The second is that some formulas, such as the one for the argument of an
$L$
-function in Theorem 4 below, are surprisingly explicit.
The contents of the paper fall into two parts. The first begins in §2, where we give some background on quadratic characters and
$L$
-functions and then prove a hybrid formula for
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
(Theorem 1). By this we mean a representation of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
as a product over prime polynomials times a product over its zeros. In §3, we prove that partial Euler products
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
approximate
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
well inside the disc
$|u|\leqslant q^{-1/2}$
when
$u$
is not close to any zero
$u_{j},~j=1,2,\ldots ,2g,$
of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
and that at every point in the disc except the
$u_{j}$
,
$\lim _{K\rightarrow \infty }P_{K}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. In §4, we obtain explicit expressions for
$\arg {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
and
$\arg P_{K}(u,\unicode[STIX]{x1D712}_{D})$
and bound their difference. In the following section we reprove a recent estimate for
$\arg {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
of Faifman and Rudnick [Reference Faifman and Rudnick7] and show that if
$K$
is sufficiently large, this bound holds for
$\arg P_{K}(u,\unicode[STIX]{x1D712}_{D})$
as well. We also reprove, in a slightly different way, another result from [Reference Faifman and Rudnick7], an estimate for the counting function
$N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
of the zeros of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
on the arc
$q^{-1/2}e(\unicode[STIX]{x1D719})$
,
$0\leqslant \unicode[STIX]{x1D719}\leqslant \unicode[STIX]{x1D703}\leqslant 1$
.
The second part of the paper begins with §6. We introduce an auxiliary function
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
modelled on the approximate functional equation (7), which equals
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
on the all important circle
$|u|=q^{-1/2}$
and has the same zeros as
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
in the complex plane. In §7, we construct a model
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
built from the truncated Euler products
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
. We then show that the Riemann hypothesis holds for
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
, that inside
$|u|\leqslant q^{-1/2}$
,
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
approximates
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
well when
$u$
is away from zeros
$u_{j}$
of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
and
$K$
is large enough, and that in this disc
$\lim _{K\rightarrow \infty }{\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
if
$u$
is not a
$u_{j}$
. Finally, in the eighth section we estimate the counting function
$N_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
of the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
, show that the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
cluster around the zeros of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
as
$K\rightarrow \infty$
, and show that when
$K$
is not too large, almost all the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
are simple.
In this paper, our main interest is when the cardinality
$q$
of the ground field
$\mathbb{F}_{q}$
is fixed and the genus
$g$
gets large, i.e.,
$\text{deg}D\rightarrow \infty$
. It would be interesting to know whether an analysis similar to that of the current paper can be carried out with
$g$
fixed and
$q\rightarrow \infty$
.
2 Background on
$L$
-functions and a hybrid formula for
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
For a prime polynomial
$P$
and any
$f\in \mathbb{F}_{q}[x]$
, the quadratic residue symbol
$(\frac{f}{P})$
is defined by
$$\begin{eqnarray}\biggl(\frac{f}{P}\biggr)=\left\{\begin{array}{@{}ll@{}}0 & \text{if}~P\mid f,\\ 1 & \text{if}~P\not |\,f~\text{and}~f~\text{is a square modulo}~P,\\ -1 & \text{if}~P\not |\,f~\text{and}~f~\text{is a non square modulo}~P.\\ \end{array}\right.\end{eqnarray}$$
If
$Q=P_{1}^{e_{1}}P_{2}^{e_{2}}\cdots P_{k}^{e_{k}}$
is the prime factorization of a monic polynomial
$Q\in \mathbb{F}_{q}[x]$
, the Jacobi symbol is defined as
$$\begin{eqnarray}\biggl(\frac{f}{Q}\biggr)=\mathop{\prod }_{i=1}^{k}\biggl(\frac{f}{P_{i}}\biggr)^{e_{i}}.\end{eqnarray}$$
If
$c\in \mathbb{F}_{q}^{\ast }$
, then
$$\begin{eqnarray}\biggl(\frac{c}{Q}\biggr)=c^{((q-1)/2)\,\text{deg}\,Q}.\end{eqnarray}$$
If
$A$
and
$B$
in
$\in \mathbb{F}_{q}[x]$
are monic coprime polynomials, the quadratic reciprocity law, proved by Artin, says that
$$\begin{eqnarray}\biggl(\frac{A}{B}\biggr)=\biggl(\frac{B}{A}\biggr)(-1)^{((q-1)/2)\text{deg}\,A\,\text{deg}\,B}.\end{eqnarray}$$
This also holds for
$A,B$
not coprime as then both sides equal zero.
For
$D\in \mathbb{F}_{q}[x]$
monic and square-free, we define the quadratic character
$\unicode[STIX]{x1D712}_{D}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{D}(f)=\biggl(\frac{D}{f}\biggr).\end{eqnarray}$$
For each such character, there corresponds an
$L$
-function (see (1)–(5) above)
$$\begin{eqnarray}L(s,\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{f\,\text{monic}}\frac{\unicode[STIX]{x1D712}_{D}(f)}{|f|^{s}}=\mathop{\prod }_{P\,\text{prime}}(1-\unicode[STIX]{x1D712}_{D}(P)|P|^{-s})^{-1},\quad \operatorname{Re}s>1.\end{eqnarray}$$
For each
$D$
in the hyperelliptic ensemble
$$\begin{eqnarray}{\mathcal{H}}_{2g+1,q}=\{D\in \mathbb{F}_{q}[x]:D\text{ monic and square-free},\deg D=2g+1\},\end{eqnarray}$$
there is an associated hyperelliptic curve given in affine form by
$$\begin{eqnarray}C_{D}:y^{2}=D(x).\end{eqnarray}$$
These curves are non-singular and of genus
$g$
, and the
$L$
-function defined above is related to the zeta function of the curve
$C_{D}$
as follows. Recall that if
$C$
is a smooth, projective, connected curve of genus
$g$
over
$\mathbb{F}_{q}$
, its zeta function is defined as
$$\begin{eqnarray}Z_{C}(u)=\exp \bigg(\mathop{\sum }_{r=1}^{\infty }N_{r}(C)\frac{u^{r}}{r}\bigg),\end{eqnarray}$$
where
$N_{r}(C)$
is the number of points on
$C$
with coordinates in
$\mathbb{F}_{q^{r}}$
(including the point at infinity). Weil [Reference Weil17] proved that
$$\begin{eqnarray}Z_{C}(u)=\frac{P_{C}(u)}{(1-u)(1-qu)},\end{eqnarray}$$
where
$P_{C}(u)$
is a polynomial of degree
$2g$
, and he proved the Riemann hypothesis for
$Z_{C}(u)$
, which states that all the zeros of
$P_{C}(u)$
lie on the circle
$|u|=q^{-1/2}$
. In the case of our hyperelliptic curves
$C_{D}$
of odd degree, it turns out that the polynomial
$P_{C_{D}}(u)$
is exactly
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
(this was first shown by Artin [Reference Artin1]). As was mentioned above, we may therefore write
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=\mathop{\prod }_{j=1}^{2g}(1-\unicode[STIX]{x1D6FC}_{j}u),\quad u\in \mathbb{C},\end{eqnarray}$$
where the
$\unicode[STIX]{x1D6FC}_{j}=q^{1/2}e(-\unicode[STIX]{x1D703}_{j}),j=1,2,\ldots ,2g,$
are the reciprocals of the roots
$u_{j}=q^{-1/2}e(\unicode[STIX]{x1D703}_{j})$
of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
.
For a monic polynomial
$f$
, we write
$\unicode[STIX]{x1D6EC}(f)=\deg P$
if
$f=P^{k}$
for some prime
$P$
and positive integer
$k$
, and
$\unicode[STIX]{x1D6EC}(f)=0$
otherwise. The logarithmic derivative of (3) may then be written
$$\begin{eqnarray}\displaystyle \frac{{\mathcal{L}}^{\prime }}{{\mathcal{L}}}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \mathop{\sum }_{P\,\text{prime}}\frac{(\text{deg}\,P)\unicode[STIX]{x1D712}_{D}(P)u^{\text{deg}\,P-1}}{1-\unicode[STIX]{x1D712}_{D}(P)u^{\text{deg}\,P}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{n=1}^{\infty }\biggl(\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=n}}\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)\biggr)u^{n-1}.\nonumber\end{eqnarray}$$
On the other hand, the logarithmic derivative of (5) is
$$\begin{eqnarray}\displaystyle \frac{{\mathcal{L}}^{\prime }}{{\mathcal{L}}}(u,\unicode[STIX]{x1D712}_{D})=-\mathop{\sum }_{n=1}^{\infty }\biggl(\mathop{\sum }_{j=1}^{2g}\unicode[STIX]{x1D6FC}_{j}^{n}\biggr)u^{n-1}. & & \displaystyle \nonumber\end{eqnarray}$$
Equating these two expressions, we find that
$$\begin{eqnarray}-\mathop{\sum }_{j=1}^{2g}e(-n\unicode[STIX]{x1D703}_{j})=\frac{1}{q^{n/2}}\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=n}}\unicode[STIX]{x1D712}_{D}(f)\unicode[STIX]{x1D6EC}(f).\end{eqnarray}$$
Using this fundamental formula, we prove a version of the hybrid formula of Gonek et al [Reference Gonek, Hughes and Keating11] (see also [Reference Bui and Keating4] and [Reference Bui and Keating5]) for
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
.
Theorem 1 (Hybrid formula for
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
).
Let
$K\geqslant 0$
be an integer and let
$$\begin{eqnarray}P_{K}(u,\unicode[STIX]{x1D712}_{D})=\exp \biggl(\mathop{\sum }_{k=1}^{K}\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=k}}\frac{\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)u^{k}}{k}\biggr),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6EC}(f)=\deg P$
if
$f=P^{n}$
for some prime polynomial
$P$
, and
$\unicode[STIX]{x1D6EC}(f)=0$
otherwise. Also, set
$$\begin{eqnarray}Z_{K}(u,\unicode[STIX]{x1D712}_{D})=\exp \bigg(\!-\!\mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k>K}\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\bigg)\bigg).\end{eqnarray}$$
Then, for
$|u|\leqslant q^{-1/2}$
,
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=P_{K}(u,\unicode[STIX]{x1D712}_{D})\,Z_{K}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
Remark 1. Bui and Florea [Reference Bui and Florea3] have, independently and at the same time as the current authors, proved a slightly different (weighted) version of the hybrid formula. They used it to calculate low moments of the
$L$
-function along the lines of Gonek et al [Reference Gonek, Hughes and Keating11].
Remark 2. One can prove similar formulas for other
$L$
-functions defined over finite fields such as for all Dirichlet
$L$
-functions
$L(s,\unicode[STIX]{x1D712})$
with
$\unicode[STIX]{x1D712}$
a Dirichlet character modulo
$Q\in \mathbb{F}_{q}[x]$
.
Remark 3. We see that
$\lim _{u\rightarrow u_{j}}Z_{K}(u,\unicode[STIX]{x1D712}_{D})=0$
within any sector
$|\text{arg}(u-u_{j})|<\unicode[STIX]{x1D70B}/2-\unicode[STIX]{x1D6FF}$
, where
$0<\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D70B}/2$
is fixed. Thus, we may interpret
$Z_{K}(u_{j},\unicode[STIX]{x1D712}_{D}),j=1,2,\ldots ,2g,$
as zero, even though the infinite series defining
$Z_{K}(u,\unicode[STIX]{x1D712}_{D})$
does not converge at
$u_{j}$
. The reader should keep this convention in mind throughout.
Remark 4. If
$K=0$
, the sum defining
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
is empty, so we interpret
$P_{0}(u,\unicode[STIX]{x1D712}_{D})$
as being identically
$1$
. Thus,
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=Z_{0}(u,\unicode[STIX]{x1D712}_{D})$
for
$|u|\leqslant q^{-1/2}$
. Indeed, for such
$u$
we see that
$$\begin{eqnarray}\displaystyle Z_{0}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \exp \bigg(\!-\!\mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k=1}^{\infty }\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\bigg)\bigg)\nonumber\\ \displaystyle & = & \displaystyle \exp \bigg(\mathop{\sum }_{j=1}^{2g}\log (1-\unicode[STIX]{x1D6FC}_{j}u)\bigg)=\mathop{\prod }_{j=1}^{2g}(1-\unicode[STIX]{x1D6FC}_{j}u).\end{eqnarray}$$
In the other direction, for
$|u|\leqslant q^{-1/2}$
with small arcs around the zeros
$u_{j}$
removed, we see from (10) that
$\lim _{K\rightarrow \infty }Z_{K}(u,\unicode[STIX]{x1D712}_{D})=1$
uniformly. Hence, on this set
$$\begin{eqnarray}\displaystyle \lim _{K\rightarrow \infty }P_{K}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \exp \bigg(\mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=k}}\frac{\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)u^{k}}{k}\bigg)\nonumber\\ \displaystyle & = & \displaystyle \exp \bigg(\mathop{\sum }_{P}\mathop{\sum }_{l=1}^{\infty }\frac{\unicode[STIX]{x1D712}_{D}(P^{l})u^{l\deg P}}{l}\bigg)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{\substack{ P\,\text{prime}}}(1-\unicode[STIX]{x1D712}_{D}(P)u^{\text{deg}\,P})^{-1}.\nonumber\end{eqnarray}$$
In other words, at the extremes,
$K=0$
and
$K=\infty$
, we (essentially) recover expressions for
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
as a product over zeros and a product over primes, respectively, from the hybrid formula.
Proof. Assume first that
$|u|<q^{-1/2}$
. Taking logarithms of both sides of (5) and using the Taylor series for
$-\text{log}(1-z),|z|<1$
, we see that
$$\begin{eqnarray}\displaystyle \log {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle -\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k=1}^{\infty }\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\biggr)=-\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k=1}^{K}+\mathop{\sum }_{k=K+1}^{\infty }\biggr)\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{k=1}^{K}\frac{u^{k}}{k}\bigg(\mathop{\sum }_{j=1}^{2g}\unicode[STIX]{x1D6FC}_{j}^{k}\bigg)-\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k=K+1}^{\infty }\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\biggr).\nonumber\\ \displaystyle & & \displaystyle \nonumber\end{eqnarray}$$
By (8), the first double sum equals
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k\leqslant K}\frac{u^{k}}{k}\biggl(\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=k}}\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
The second is simply
$$\begin{eqnarray}-\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k>K}\frac{(q^{1/2}e(-\unicode[STIX]{x1D703}_{j})u)^{k}}{k}\biggr).\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle \log {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \mathop{\sum }_{k=1}^{K}\biggl(\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=k}}\frac{\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)u^{k}}{k}\biggr)-\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k>K}\frac{(q^{1/2}e(-\unicode[STIX]{x1D703}_{j})u)^{k}}{k}\biggr).\nonumber\end{eqnarray}$$
Exponentiating this, we obtain (11) for
$|u|<q^{-1/2}$
. It only remains to treat the case
$|u|=q^{-1/2}$
. On this circle the first term in (13) is a polynomial in
$u$
, so is continuous. The second term, with
$u=q^{-1/2}e(\unicode[STIX]{x1D703})$
and
$\unicode[STIX]{x1D703}\neq \unicode[STIX]{x1D703}_{j}$
,
$j=1,2,\ldots ,2g$
, equals
$$\begin{eqnarray}-\mathop{\sum }_{j=1}^{2g}\biggl(\mathop{\sum }_{k>K}\frac{e(k(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))}{k}\biggr).\end{eqnarray}$$
By partial summation, the series
$\sum _{k>K}e(k\unicode[STIX]{x1D719})/k$
converges uniformly for
$\unicode[STIX]{x1D6FF}\leqslant \unicode[STIX]{x1D719}\leqslant 1-\unicode[STIX]{x1D6FF}$
, where
$0<\unicode[STIX]{x1D6FF}<\frac{1}{2}$
is fixed. It follows that (13) is continuous on the circle
$|u|=q^{-1/2}$
with the points
$u_{j}$
deleted. Thus,
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
and the function
$P_{K}(u,\unicode[STIX]{x1D712}_{D})\,Z_{K}(u,\unicode[STIX]{x1D712}_{D})$
agree and are analytic in
$|u|<q^{-1/2}$
, and are continuous on the circle
$|u|=q^{-1/2}$
minus the points
$u_{1},\ldots ,u_{2g}$
. They therefore agree on this set and, by our interpretation of
$Z_{K}$
as zero in the limit as
$u\rightarrow u_{j}$
, they agree at these points as well. This completes the proof of the theorem.◻
3 Approximation of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
by
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
In this section, it simplifies some expressions slightly if we use both the notations
$\log g$
and
$\log _{q}g$
.
Let
$u=q^{-\unicode[STIX]{x1D70E}-\text{i}t}$
with
$\unicode[STIX]{x1D70E}>\frac{1}{2}$
and assume that
$K\geqslant 1$
. Then
$$\begin{eqnarray}\displaystyle |\text{log}\,Z_{K}(u,\unicode[STIX]{x1D712}_{D})| & = & \displaystyle \bigg|\mathop{\sum }_{j=1}^{2g}\mathop{\sum }_{k>K}\frac{(\unicode[STIX]{x1D6FC}_{j}u)^{k}}{k}\bigg|\leqslant 2g\mathop{\sum }_{k>K}\frac{q^{(1/2-\unicode[STIX]{x1D70E})k}}{k}\leqslant \frac{2g~q^{(1/2-\unicode[STIX]{x1D70E})K}}{K(1-q^{1/2-\unicode[STIX]{x1D70E}})}.\nonumber\end{eqnarray}$$
If we write
$\unicode[STIX]{x1D70E}=\frac{1}{2}+C\log _{q}g/K$
, with
$C>0$
and possibly depending on
$g$
, then this equals
$$\begin{eqnarray}\frac{2g^{1-C}}{K(1-g^{-C/K})}.\end{eqnarray}$$
Since
$1-\text{e}^{-x}\geqslant x/2$
for
$0\leqslant x\leqslant \frac{1}{2}$
, if we assume that
$K\geqslant 2C\log g$
, we have
$$\begin{eqnarray}1-g^{-C/K}\geqslant \frac{C\log g}{2K}.\end{eqnarray}$$
Hence, the above is
$$\begin{eqnarray}{\leqslant}\frac{4g^{1-C}}{C\log g}.\end{eqnarray}$$
Using this with Theorem 1, we see that if
$C\geqslant 1$
, then
$$\begin{eqnarray}\displaystyle {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=P_{K}(u,\unicode[STIX]{x1D712}_{D})\biggl(1+O\biggl(\frac{1}{Cg^{C-1}\log g}\biggr)\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Theorem 2. Let
$\unicode[STIX]{x1D70E}=\frac{1}{2}+(C\log _{q}g)/K$
with
$C\geqslant 1$
(recall that
$C$
might depend on
$g$
) and
$K\geqslant 2C\log g$
. Then, for
$|u|\leqslant q^{-\unicode[STIX]{x1D70E}}$
, we have
$$\begin{eqnarray}\displaystyle {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=P_{K}(u,\unicode[STIX]{x1D712}_{D})\biggl(1+O\biggl(\frac{1}{Cg^{C-1}\log g}\biggr)\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
We can prove a similar approximation on the circle
$|u|=q^{-1/2}$
. Write
$u=q^{-1/2}e(\unicode[STIX]{x1D703})$
. Then, by (10),
$$\begin{eqnarray}\displaystyle & \displaystyle Z_{K}(e(\unicode[STIX]{x1D703})q^{-1/2},\unicode[STIX]{x1D712}_{D})=\exp \bigg(\!-\!\mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k>K}\frac{e(k(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))}{k}\bigg)\bigg). & \displaystyle \nonumber\end{eqnarray}$$
By partial summation,
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k>K}\frac{e(k(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))}{k}\bigg)\ll \frac{1}{K+1}\mathop{\sum }_{j=1}^{2g}\frac{1}{|\text{sin}(\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))|}. & \displaystyle \nonumber\end{eqnarray}$$
Let
$\Vert x\Vert =\min _{n\in \mathbf{Z}}|x-n|$
. Then, if we assume that
$$\begin{eqnarray}\min _{1\leqslant j\leqslant 2g}\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert \geqslant \frac{c}{2g}\end{eqnarray}$$
and use the estimate
$|\text{sin}\,\unicode[STIX]{x1D70B}x|\geqslant 2\Vert x\Vert$
, we find that the above is
$$\begin{eqnarray}\ll \frac{g^{2}}{cK}.\end{eqnarray}$$
Thus, we have the following result.
Theorem 3. Let
$u=q^{-1/2}e(\unicode[STIX]{x1D703})$
. Suppose that
$c>0$
,
$$\begin{eqnarray}\min _{1\leqslant j\leqslant 2g}\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert \geqslant \frac{c}{2g},\end{eqnarray}$$
and
$K\geqslant g^{2}/c.$
Then
$$\begin{eqnarray}\displaystyle {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=P_{K}(u,\unicode[STIX]{x1D712}_{D})\biggl(1+O\biggl(\frac{g^{2}}{cK}\biggr)\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Observe that as a consequence of Theorems 2 and 3, we have for
$|u|\leqslant q^{-1/2}$
,
$u\neq u_{j},j=1,2,\ldots ,2g$
, that
$$\begin{eqnarray}\lim _{K\rightarrow \infty }P_{K}(u,\unicode[STIX]{x1D712}_{D})\rightarrow {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
This was pointed out in Remark 3 above, but Theorems 2 and 3 also supply rates of convergence.
4 Explicit expressions for
$\arg {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
and
$\arg P_{K}(u,\unicode[STIX]{x1D712}_{D})$
A standard way to define the argument of a quadratic Dirichlet
$L$
-function
$L(\frac{1}{2}+\text{i}t,\unicode[STIX]{x1D712}_{D})$
when
$t$
is not the ordinate of a zero is by continuous variation starting with the value zero at
$s=2$
, moving up the line
$\unicode[STIX]{x1D70E}=2$
as far as
$2+\text{i}t,$
and then horizontally over to
$s=\frac{1}{2}+\text{i}t$
. This makes sense for our function field
$L$
-functions defined in (1) and (2) as well. For the alternative form
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
of
$L(s,\unicode[STIX]{x1D712}_{D})$
, however, this corresponds to continuous variation in the negative sense along the circular arc from
$q^{-2}$
to a point
$q^{-2}e(-\unicode[STIX]{x1D703})$
and then along the radius
$re(-\unicode[STIX]{x1D703})$
from
$r=q^{-2}$
to
$r=q^{-1/2}e(-\unicode[STIX]{x1D703}).$
This would be
$\arg {\mathcal{L}}(q^{-1/2}e(-\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
, which is the same as
$-\text{arg}\,{\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
, since
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
is real for real values of
$u$
. Thus, denoting the path consisting of the positively oriented circular arc from
$q^{-2}$
to
$q^{-2}e(\unicode[STIX]{x1D703})$
followed by the radial segment from
$q^{-2}e(\unicode[STIX]{x1D703})$
to
$q^{-1/2}e(\unicode[STIX]{x1D703})$
by
$\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})$
, we define
$$\begin{eqnarray}\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})=-\triangle _{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
If
$q^{-1/2}e(\unicode[STIX]{x1D703})$
happens to be a zero of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
, we use the convention that
$$\begin{eqnarray}\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})=\lim _{\unicode[STIX]{x1D716}\rightarrow 0^{+}}\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}+\unicode[STIX]{x1D716}),\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
We also define
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\frac{1}{\unicode[STIX]{x1D70B}}\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
Similarly, we let
$$\begin{eqnarray}\arg P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})=-\triangle _{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})}P_{K}(u,\unicode[STIX]{x1D712}_{D})\end{eqnarray}$$
and
$$\begin{eqnarray}S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\frac{1}{\unicode[STIX]{x1D70B}}\arg P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
Our next goal is to obtain alternative expressions for these arguments. From (5) and (14), we find that if
$\unicode[STIX]{x1D703}\neq \unicode[STIX]{x1D703}_{j}$
for any
$j=1,2,\ldots ,2g$
, then
$$\begin{eqnarray}\displaystyle \arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}) & = & \displaystyle -\triangle _{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})}\arg \mathop{\prod }_{j=1}^{2g}(1-\unicode[STIX]{x1D6FC}_{j}u)\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{j=1}^{2g}\triangle _{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})}\arg (1-e(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})),\nonumber\end{eqnarray}$$
where, on the last line, we use the value of the argument in
$(-\unicode[STIX]{x1D70B}/2,\unicode[STIX]{x1D70B}/2)$
. Elementary geometric reasoning shows that if
$\unicode[STIX]{x1D719}\notin \mathbb{Z}$
, then
$\arg (1-e(\unicode[STIX]{x1D719}))=\unicode[STIX]{x1D70B}(\{\unicode[STIX]{x1D719}\}-\frac{1}{2})$
, where
$\{x\}$
denotes the fractional part of the real number
$x$
. Thus,
$$\begin{eqnarray}\displaystyle \arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \unicode[STIX]{x1D70B}\mathop{\sum }_{j=1}^{2g}\biggl(\biggl(\{-\unicode[STIX]{x1D703}_{j}\}-\frac{1}{2}\biggr)-\biggl(\{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\}-\frac{1}{2}\biggr)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70B}\mathop{\sum }_{j=1}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\}).\nonumber\end{eqnarray}$$
It follows that for
$\unicode[STIX]{x1D703}\neq \unicode[STIX]{x1D703}_{j}$
,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{j=1}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\}).\end{eqnarray}$$
If
$\unicode[STIX]{x1D703}$
does equal
$\unicode[STIX]{x1D703}_{i}$
for some
$i$
, then, by (15) and (16),
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703}_{i},\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \lim _{\unicode[STIX]{x1D716}\rightarrow 0^{+}}\biggl(\mathop{\sum }_{j=1}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}_{i}+\unicode[STIX]{x1D716}-\unicode[STIX]{x1D703}_{j}\})\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ j=1 \\ j\neq i}}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}_{i}-\unicode[STIX]{x1D703}_{j}\})+\lim _{\unicode[STIX]{x1D716}\rightarrow 0^{+}}(\{-\unicode[STIX]{x1D703}_{i}\}-\{\unicode[STIX]{x1D703}_{i}+\unicode[STIX]{x1D716}-\unicode[STIX]{x1D703}_{i}\})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ j=1}}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}_{i}-\unicode[STIX]{x1D703}_{j}\}).\nonumber\end{eqnarray}$$
Thus, (18) holds whether or not
$\unicode[STIX]{x1D703}$
is a
$\unicode[STIX]{x1D703}_{i}$
.
We can express the formula in both cases in a unified way by using the function
$$\begin{eqnarray}\displaystyle s(x)=\left\{\begin{array}{@{}ll@{}}\{x\}-\frac{1}{2}\quad & ifx\in \mathbb{R}\setminus \mathbb{Z},\\ 0\quad & ifx\in \mathbb{ Z}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
We then clearly have for all
$\unicode[STIX]{x1D703}$
that
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{\substack{ j=1}}^{2g}(s(-\unicode[STIX]{x1D703}_{j})-s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})). & & \displaystyle\end{eqnarray}$$
Notice that since
$0<\unicode[STIX]{x1D703}_{j}<1$
for all
$j=1,2,\ldots ,2g$
and, since
$\unicode[STIX]{x1D703}_{g+j}=1-\unicode[STIX]{x1D703}_{j}$
for
$j=1,2,\ldots ,g$
, we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ j=1}}^{2g}s(-\unicode[STIX]{x1D703}_{j}) & = & \displaystyle \mathop{\sum }_{\substack{ j=1}}^{2g}\biggl(\{-\unicode[STIX]{x1D703}_{j}\}-\frac{1}{2}\biggr)=\mathop{\sum }_{\substack{ j=1}}^{2g}\biggl((1-\unicode[STIX]{x1D703}_{j})-\frac{1}{2}\biggr)\nonumber\\ \displaystyle & = & \displaystyle g-\mathop{\sum }_{\substack{ j=1}}^{2g}\unicode[STIX]{x1D703}_{j}=g-\mathop{\sum }_{\substack{ j=1}}^{g}(\unicode[STIX]{x1D703}_{j}+(1-\unicode[STIX]{x1D703}_{j}))=0.\nonumber\end{eqnarray}$$
Thus, (19) is equivalent to
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=-\mathop{\sum }_{\substack{ j=1}}^{2g}s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}). & & \displaystyle \nonumber\end{eqnarray}$$
Now it is well known (for example, see Montgomery and Vaughan [Reference Montgomery and Vaughan13, p. 536]) that
$$\begin{eqnarray}\displaystyle s(x)=-\mathop{\sum }_{k=1}^{K}\frac{\sin (2\unicode[STIX]{x1D70B}xk)}{\unicode[STIX]{x1D70B}k}+E_{K}(x), & & \displaystyle\end{eqnarray}$$
where, for
$K\geqslant 1$
,
$$\begin{eqnarray}|E_{K}(x)|\leqslant \min \bigg(\frac{1}{2},\frac{1}{\unicode[STIX]{x1D70B}(2K+1)|\text{sin}\,\unicode[STIX]{x1D70B}x|}\bigg)\leqslant \min \bigg(\frac{1}{2},\frac{1}{4\unicode[STIX]{x1D70B}K\Vert x\Vert }\bigg).\end{eqnarray}$$
From this bound, we see that the series
$$\begin{eqnarray}-\!\mathop{\sum }_{k=1}^{\infty }\frac{\sin (2\unicode[STIX]{x1D70B}xk)}{\unicode[STIX]{x1D70B}k}\end{eqnarray}$$
converges pointwise to
$s(x)$
when
$x\notin \mathbb{Z}$
. Moreover, the series clearly converges to
$s(x)$
when
$x\in \mathbb{Z}$
as well, since then every term is zero. We may therefore summarize the above in the following result.
Theorem 4. For
$\unicode[STIX]{x1D703}\in \mathbb{R}$
,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{\substack{ j=1}}^{2g}(s(-\unicode[STIX]{x1D703}_{j})-s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))=-\mathop{\sum }_{\substack{ j=1}}^{2g}s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{j=1}^{2g}\mathop{\sum }_{k=1}^{\infty }\frac{\sin (2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})k)}{\unicode[STIX]{x1D70B}k}. & & \displaystyle \nonumber\end{eqnarray}$$
Note that the second formula in the theorem is what we would obtain formally from the first line of (12) on taking (14) and basic properties of the
$\unicode[STIX]{x1D703}_{j}$
into account.
We can obtain similar expressions for
$\arg P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
. From (9) and (17),
$$\begin{eqnarray}\displaystyle \arg P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}) & = & \displaystyle -\triangle _{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})}\;P_{K}(u,\unicode[STIX]{x1D712}_{D})\nonumber\\ \displaystyle & = & \displaystyle -\text{Im}\;\mathop{\sum }_{k=1}^{K}\;\frac{e(k\unicode[STIX]{x1D703})}{k}\biggl(q^{-k/2}\mathop{\sum }_{\substack{ f\,\text{monic} \\ \deg f=k}}\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)\biggr).\nonumber\end{eqnarray}$$
Using (8) to replace the expression in parentheses, we find that
$$\begin{eqnarray}\displaystyle \arg P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \operatorname{Im}\;\mathop{\sum }_{k=1}^{K}\;\frac{e(k\unicode[STIX]{x1D703})}{k}\bigg(\mathop{\sum }_{j=1}^{2g}e(-k\unicode[STIX]{x1D703}_{j})\bigg)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k=1}^{K}\;\frac{\sin (2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})k)}{k}\bigg).\nonumber\end{eqnarray}$$
By (20), this equals
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70B}\mathop{\sum }_{j=1}^{2g}(-s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})+E_{K}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})). & \displaystyle \nonumber\end{eqnarray}$$
Hence, we have the following result.
Theorem 5. For
$K\geqslant 1$
,
$$\begin{eqnarray}S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{j=1}^{2g}(-s(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})+E_{K}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))\end{eqnarray}$$
and
$$\begin{eqnarray}S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k=1}^{K}\frac{\sin (2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})k)}{\unicode[STIX]{x1D70B}k}\bigg),\end{eqnarray}$$
where
$$\begin{eqnarray}|E_{K}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})|\leqslant \min \bigg(\frac{1}{2},\frac{1}{4\unicode[STIX]{x1D70B}K\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert }\bigg).\end{eqnarray}$$
From Theorems 4 and 5, we immediately have the following result.
Corollary 6. For
$K\geqslant 1$
,
$$\begin{eqnarray}|S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})|\leqslant \mathop{\sum }_{j=1}^{2g}\min \bigg(\frac{1}{2},\frac{1}{4\unicode[STIX]{x1D70B}K\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert }\bigg).\end{eqnarray}$$
5 The counting function for the zeros of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
It is a simple matter to count the number of zeros of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
on an arc of the circle
$|u|=q^{-1/2}$
. This was done by a slightly different method by Faifman and Rudnick [Reference Faifman and Rudnick7] for the hyperelliptic ensemble
${\mathcal{H}}_{2g+2,q}$
of even degree monic polynomials. We include a proof because it is short.
Let
$N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
denote the number of zeros of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
on the circular arc
$q^{-1/2}e(\unicode[STIX]{x1D719})$
,
$0\leqslant \unicode[STIX]{x1D719}\leqslant \unicode[STIX]{x1D703}\leqslant 1$
. That is,
$$\begin{eqnarray}\displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{\unicode[STIX]{x1D703}_{j}\leqslant \unicode[STIX]{x1D703}}1. & & \displaystyle \nonumber\end{eqnarray}$$
For the moment we assume that
$\unicode[STIX]{x1D703}\neq \unicode[STIX]{x1D703}_{j},j=1,2,\ldots ,2g$
. Let
$C(\unicode[STIX]{x1D703})$
be the positively oriented contour consisting of the circular arc
$qe(\unicode[STIX]{x1D719})$
from
$\unicode[STIX]{x1D719}=0$
to
$\unicode[STIX]{x1D719}=\unicode[STIX]{x1D703}$
, the radial segment
$re(\unicode[STIX]{x1D703})$
from
$r=q$
to
$r=q^{-2}$
, the circular arc
$e(\unicode[STIX]{x1D719})q^{-2}$
from
$\unicode[STIX]{x1D719}=\unicode[STIX]{x1D703}$
to
$\unicode[STIX]{x1D719}=0$
, and then the real segment from
$r=q^{-2}$
to
$r=q$
. Then
$$\begin{eqnarray}N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\frac{1}{2\unicode[STIX]{x1D70B}}\triangle _{C}\arg {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
The change in argument along the real segment is zero. Along the outer circular arc from
$q$
to
$qe(\unicode[STIX]{x1D703})$
, and then along the radius from
$qe(\unicode[STIX]{x1D703})$
to
$q^{-1/2}e(\unicode[STIX]{x1D703})$
, we use the functional equation
$$\begin{eqnarray}\displaystyle {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=(qu^{2})^{g}{\mathcal{L}}(1/qu,\unicode[STIX]{x1D712}_{D}) & & \displaystyle \nonumber\end{eqnarray}$$
to see that the change in argument equals
$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}g+\triangle _{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703})}\arg {\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
, where
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703})$
is the contour consisting of the circular arc from
$u=q^{-2}$
to
$u=q^{-2}e(-\unicode[STIX]{x1D703})$
and then continuing along the radius
$re(-\unicode[STIX]{x1D703})$
from
$r=q^{-2}$
to
$q^{-1/2}.$
But this is just minus the change in argument along
$\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703})$
(see just above (14)), which we defined to be
$\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
. Thus, the change along the outer part of the contour equals
$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}g+\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
. By (14), the remaining change in argument also equals
$\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})$
. Thus,
$$\begin{eqnarray}\displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D70B}}(4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}g+2\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D}))\nonumber\\ \displaystyle & = & \displaystyle 2g\unicode[STIX]{x1D703}+\frac{1}{\unicode[STIX]{x1D70B}}\arg {\mathcal{L}}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})\nonumber\\ \displaystyle & = & \displaystyle 2g\unicode[STIX]{x1D703}+S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
If
$\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{j}$
for some
$j$
, our convention (15) means that (23) holds in this case as well. Thus, we have proved the following result.
Theorem 7. For all
$\unicode[STIX]{x1D703}\in [0,1]$
, we have
$$\begin{eqnarray}N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=2g\unicode[STIX]{x1D703}+S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
As a check of this formula, we perform the following calculation. According to Theorem 4, if
$0\leqslant \unicode[STIX]{x1D703}\leqslant 1$
,
$$\begin{eqnarray}\displaystyle & \displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\mathop{\sum }_{\substack{ j=1}}^{2g}(\{-\unicode[STIX]{x1D703}_{j}\}-\{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\}). & \displaystyle \nonumber\end{eqnarray}$$
Now
$\{-\unicode[STIX]{x1D703}_{j}\}=1-\unicode[STIX]{x1D703}_{j}$
and
$$\begin{eqnarray}\{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\quad & \text{if }0<\unicode[STIX]{x1D703}_{j}\leqslant \unicode[STIX]{x1D703},\\ 1+\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\quad & \text{if }\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{j}<1.\end{array}\right.\end{eqnarray}$$
Hence, the above is
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \mathop{\sum }_{0<\unicode[STIX]{x1D703}_{j}\leqslant \unicode[STIX]{x1D703}}((1-\unicode[STIX]{x1D703}_{j})-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))+\mathop{\sum }_{\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{j}<1}((1-\unicode[STIX]{x1D703}_{j})-(1+\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{0<\unicode[STIX]{x1D703}_{j}\leqslant \unicode[STIX]{x1D703}}(1-\unicode[STIX]{x1D703})-\mathop{\sum }_{\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{j}<1}\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & = & \displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-2g\unicode[STIX]{x1D703},\nonumber\end{eqnarray}$$
which agrees with (24).
6 Upper bounds for
$S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
and
$S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
Faifman [Reference Faifman6] (see also Faifman and Rudnick [Reference Faifman and Rudnick7, Proposition 5.1]) has shown that for the hyperelliptic ensemble
${\mathcal{H}}_{2g+2,q}$
,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \frac{g}{\log _{q}g}.\end{eqnarray}$$
This is the analogue of the best known bound for the order of
$S(t)=(1/\unicode[STIX]{x1D70B})\arg \unicode[STIX]{x1D701}(\frac{1}{2}+\text{i}t)$
on the Riemann hypothesis, namely,
$$\begin{eqnarray}S(t)\ll \frac{\log t}{\log \log t}.\end{eqnarray}$$
It is clear that the methods of [Reference Faifman6] and [Reference Faifman and Rudnick7] apply to our ensemble
${\mathcal{H}}_{2g+1,q}$
as well. In this section, we first give a proof of this and then show that the same bound holds for
$S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
if
$K$
is sufficiently large with respect to
$g$
.
We use the following approximation result, which we state without proof (see, for example, Montgomery [Reference Montgomery12]).
Lemma 8. Let
$I=[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}]$
be an arc in
$\mathbb{T}$
with length
$\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}<1$
. Then, for any positive integer
$K$
, there are trigonometric polynomials
$$\begin{eqnarray}T^{\pm }(x)=\mathop{\sum }_{k=-K}^{K}a^{\pm }(k)e(kx)\end{eqnarray}$$
such that:
-
(a)
$T^{-}(x)\leqslant \unicode[STIX]{x1D712}_{I}(x)\leqslant T^{+}(x)$
for all
$x$
; -
(b)
$\int _{0}^{1}T^{\pm }(x)\,dx=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}\pm 1/(K+1)$
.
Theorem 9. For
$0<\unicode[STIX]{x1D703}<1$
,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \frac{g}{\log _{q}g}.\end{eqnarray}$$
Proof. For
$0<\unicode[STIX]{x1D703}<1$
, we have
$$\begin{eqnarray}\displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \mathop{\sum }_{j=1}^{2g}\unicode[STIX]{x1D712}_{[0,\unicode[STIX]{x1D703}]}(\unicode[STIX]{x1D703}_{j})\leqslant \mathop{\sum }_{j=1}^{2g}T^{+}(\unicode[STIX]{x1D703}_{j})\nonumber\\ \displaystyle & = & \displaystyle 2ga^{+}(0)\,+\,\mathop{\sum }_{\substack{ k=-K \\ k\neq 0}}^{K}a^{+}(k)\bigg(\mathop{\sum }_{j=1}^{2g}e(k\unicode[STIX]{x1D703}_{j})\bigg).\nonumber\end{eqnarray}$$
By (8) and part (b) of the lemma, we thus see that
$$\begin{eqnarray}\displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\leqslant 2g\biggl(\unicode[STIX]{x1D703}+\frac{1}{K+1}\biggr)-2\mathop{\sum }_{k=1}^{K}\frac{a^{+}(k)}{q^{k/2}}\bigg(\mathop{\sum }_{\deg f=k}\unicode[STIX]{x1D6EC}(f)\unicode[STIX]{x1D712}_{D}(f)\bigg). & & \displaystyle\end{eqnarray}$$
Recall that if
$f\in L^{1}(\mathbb{T})$
, then
$$\begin{eqnarray}|\hat{f}(k)|=\bigg|\int _{0}^{1}f(x)e(-kx)\,dx\bigg|\leqslant \int _{0}^{1}|f(x)|\,dx.\end{eqnarray}$$
Since
$$\begin{eqnarray}\hat{\unicode[STIX]{x1D712}}_{[0,\unicode[STIX]{x1D703}]}(k)=e(-k\unicode[STIX]{x1D703}/2)\frac{\sin \unicode[STIX]{x1D70B}k\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D70B}k}\end{eqnarray}$$
for
$k\neq 0$
, if we take
$f=T^{+}-\unicode[STIX]{x1D712}_{[0,\unicode[STIX]{x1D703}]}$
, then again by (b) of the lemma,
$$\begin{eqnarray}\displaystyle \biggl|e(-k\unicode[STIX]{x1D703}/2)\frac{\sin \unicode[STIX]{x1D70B}k\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D70B}k}-a^{+}(k)\biggr|\leqslant \int _{0}^{1}|\unicode[STIX]{x1D712}_{[0,\unicode[STIX]{x1D703}]}(x)-T^{+}(x)|\,dx\leqslant \frac{1}{K+1}. & & \displaystyle \nonumber\end{eqnarray}$$
Thus,
$$\begin{eqnarray}|a_{k}^{+}|\leqslant \frac{1}{K+1}+\bigg|\frac{\sin \unicode[STIX]{x1D70B}k\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D70B}k}\bigg|\ll \min \biggl(\Vert \unicode[STIX]{x1D703}\Vert ,\frac{1}{k}\biggr).\end{eqnarray}$$
From (26), it now follows that
$$\begin{eqnarray}\displaystyle S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}) & = & \displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-2g\unicode[STIX]{x1D703}\leqslant \frac{2g}{K}\,+O\bigg(\mathop{\sum }_{k=1}^{K}\frac{\min (\Vert \unicode[STIX]{x1D703}\Vert ,k^{-1})}{q^{k/2}}\bigg(\mathop{\sum }_{\deg f=k}\unicode[STIX]{x1D6EC}(f)\bigg)\!\bigg).\nonumber\end{eqnarray}$$
By the prime polynomial theorem, the sum in parentheses is equal to
$q^{k}$
. Thus, the second term on the right is
$$\begin{eqnarray}\ll \mathop{\sum }_{k=1}^{K}\frac{q^{k/2}}{k^{2}}\ll q^{K/2}.\end{eqnarray}$$
Hence,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\leqslant \frac{2g}{K}+O(q^{K/2}).\end{eqnarray}$$
The same argument using
$T^{-}(x)$
instead of
$T^{+}(x)$
leads to
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\geqslant -\frac{2g}{K}+O(q^{K/2}).\end{eqnarray}$$
Thus,
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \frac{g}{K}+q^{K/2}.\end{eqnarray}$$
Taking
$K=\log _{q}g$
, we obtain (25).◻
In the case of the zeta function we expect much more to be true, and the heuristic arguments in Farmer et al [Reference Farmer, Gonek and Hughes8] that indicate this also suggest that
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=O(\sqrt{g\log g})\quad and\quad S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=\unicode[STIX]{x1D6FA}(\sqrt{g\log g}).\end{eqnarray}$$
To accommodate any eventual improvements in the estimate, we state the next result in terms of a general upper bound
$\unicode[STIX]{x1D6F7}(g)$
for
$S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
.
Theorem 10. Suppose that
$$\begin{eqnarray}S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \unicode[STIX]{x1D6F7}(g).\end{eqnarray}$$
Then, for
$K\geqslant (g\log g)/\unicode[STIX]{x1D6F7}(g)$
, we have
$$\begin{eqnarray}S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \unicode[STIX]{x1D6F7}(g).\end{eqnarray}$$
In particular,
$$\begin{eqnarray}\displaystyle & \displaystyle S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\ll \frac{g}{\log _{q}g} & \displaystyle \nonumber\end{eqnarray}$$
when
$K\geqslant \log _{q}g\cdot \log g$
.
Proof. Set
$\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6F7}(g)/g$
. The right-hand side of (22) is
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m=0}^{[2g/\unicode[STIX]{x1D6E5}]+1}\mathop{\sum }_{\substack{ j \\ m\unicode[STIX]{x1D6E5}\leqslant \Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert <(m+1)\unicode[STIX]{x1D6E5}}}\min \bigg(\frac{1}{2},\frac{1}{4\unicode[STIX]{x1D70B}K\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert }\bigg). & \displaystyle \nonumber\end{eqnarray}$$
By (23),
$N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})=2g\unicode[STIX]{x1D703}+O(\unicode[STIX]{x1D6F7}(g))$
. Hence, for each
$m$
the number of terms in the inner sum over
$j$
is
$2g\unicode[STIX]{x1D6E5}+O(\unicode[STIX]{x1D6F7}(g))\ll \unicode[STIX]{x1D6F7}(g)$
. The
$m=0$
term therefore contributes
$\ll \frac{1}{2}\unicode[STIX]{x1D6F7}(g)$
and the remaining terms contribute
$$\begin{eqnarray}\displaystyle & \displaystyle \ll \frac{1}{K}\mathop{\sum }_{m=1}^{[2g/\unicode[STIX]{x1D6E5}]+1}\frac{\unicode[STIX]{x1D6F7}(g)}{m\unicode[STIX]{x1D6E5}}\ll \frac{\unicode[STIX]{x1D6F7}(g)\log (2g/\unicode[STIX]{x1D6E5})}{K\unicode[STIX]{x1D6E5}}=\frac{g\log g}{K}. & \displaystyle \nonumber\end{eqnarray}$$
Combining these estimates and taking
$K\geqslant g\log g/\unicode[STIX]{x1D6F7}(g)$
, we obtain (27). The last assertion of the theorem is an immediate consequence of (27) and (25).◻
7 Discussion of a function related to
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
We now introduce an auxiliary function
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
in order to study
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. For
$u\in \mathbb{C}$
, we define
$$\begin{eqnarray}{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\textstyle \frac{1}{2}}({\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})+(qu^{2})^{g}{\mathcal{L}}(\overline{u},\unicode[STIX]{x1D712}_{D})).\end{eqnarray}$$
Note that
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
is not holomorphic although it is harmonic.
The reason we introduce
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
is that, unlike
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
itself, we can model
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
on the closed disc
$|u|\leqslant q^{-1/2}$
along the lines of the approximate functional equation (7), using the truncated Euler products
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
. By Theorems 2 and 3,
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
is a good approximation of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
when
$K$
is large provided we are not too close to a zero of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. This is inevitable because
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
can never vanish. Indeed, Theorem 3 indicates that the closer one is to a zero, the larger
$K$
must be to retain a good approximation. Thus, the approximation of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
by
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
is least helpful where we most need it—at the zeros.
Fortunately, “knowing”
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
is in many ways the same thing as “knowing”
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. For example, on the circle
$|u|=q^{-1/2}$
,
$$\begin{eqnarray}{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
To see this, observe that
$1/qu=\overline{u}$
when
$|u|=q^{-1/2}$
and that, by the functional equation,
$$\begin{eqnarray}{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})=(qu^{2})^{g}{\mathcal{L}}(1/qu,\unicode[STIX]{x1D712}_{D})=(qu^{2})^{g}{\mathcal{L}}(\overline{u},\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
Using this in (28), we obtain (29).
As another example, consider the size of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. From (28), it is immediate that
$$\begin{eqnarray}\sup _{|u|\leqslant q^{-1/2}}|{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})|\leqslant \sup _{|u|\leqslant q^{-1/2}}|{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})|.\end{eqnarray}$$
In fact, however, the two quantities are equal. For
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
, being harmonic, must attain its maximum modulus on the disc on the boundary. However,
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
on the boundary, so
$$\begin{eqnarray}\sup _{|u|\leqslant q^{-1/2}}|{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})|=\sup _{|u|\leqslant q^{-1/2}}|{\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})|.\end{eqnarray}$$
As a final example, we prove the following result.
Theorem 11. The functions
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
and
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
have the same zeros in
$\mathbb{C}$
. In particular, the Riemann hypothesis holds for
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
.
Proof. Since
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
, on
$|u|=q^{-1/2}$
, both functions have the same zeros on this circle. Since
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
has no zeros anywhere else, to complete the proof we must show that neither does
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
.
Suppose, on the contrary, that
$u_{0}$
is a zero of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
with
$|u_{0}|\neq q^{-1/2}$
. Since
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
has no zeros off the circle
$|u|=q^{-1/2}$
,
$u_{0}$
is not a zero of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
. We may therefore write
$$\begin{eqnarray}0={\mathcal{F}}(u_{0},\unicode[STIX]{x1D712}_{D})=\frac{1}{2}{\mathcal{L}}(u_{0},\unicode[STIX]{x1D712}_{D})\bigg(1+(qu_{0}^{2})^{g}\frac{{\mathcal{L}}(\overline{u}_{0},\unicode[STIX]{x1D712}_{D})}{{\mathcal{L}}(u_{0},\unicode[STIX]{x1D712}_{D})}\bigg).\end{eqnarray}$$
The only way the term in parentheses on the right can vanish is if
$$\begin{eqnarray}\bigg|(qu_{0}^{2})^{g}\frac{{\mathcal{L}}(\overline{u}_{0},\unicode[STIX]{x1D712}_{D})}{{\mathcal{L}}(u_{0},\unicode[STIX]{x1D712}_{D})}\bigg|=1.\end{eqnarray}$$
Now
$|{\mathcal{L}}(\overline{u}_{0},\unicode[STIX]{x1D712}_{D})/{\mathcal{L}}(u_{0},\unicode[STIX]{x1D712}_{D})|=1$
, so this implies that
$|u_{0}|=q^{-1/2}$
, which is a contradiction.◻
8 A model of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
Having shown that we can deduce information about
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
from information about
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
, we now model
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
using the Euler product truncations
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
. We set
$$\begin{eqnarray}\displaystyle {\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})={\textstyle \frac{1}{2}}(P_{K}(u,\unicode[STIX]{x1D712}_{D})+(qu^{2})^{g}P_{K}(\overline{u},\unicode[STIX]{x1D712}_{D})), & & \displaystyle \nonumber\end{eqnarray}$$
where
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
is defined in (9). Since
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
has no zeros, we see that
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})=0$
if and only if
$$\begin{eqnarray}1+(qu^{2})^{g}\frac{P_{K}(\overline{u},\unicode[STIX]{x1D712}_{D})}{P_{K}(u,\unicode[STIX]{x1D712}_{D})}=0.\end{eqnarray}$$
Since
$|P_{K}(\overline{u},\unicode[STIX]{x1D712}_{D})/P_{K}(u,\unicode[STIX]{x1D712}_{D})|=1$
, this implies that
$|u|=q^{-1/2}$
. Thus, we have proved the following result.
Theorem 12 (The Riemann hypothesis for
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
).
All zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
lie on
$|u|=q^{-1/2}$
.
As
$P_{K}(u,\unicode[STIX]{x1D712}_{D})$
approximates
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
well in
$|u|\leqslant q^{-1/2}$
when
$K$
is large and
$u$
is not too close to a zero of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
,
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
approximates
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
.
Theorem 13. Let
$\unicode[STIX]{x1D70E}=\frac{1}{2}+(C\log _{q}g)/K$
with
$C\geqslant 1$
and
$K\geqslant 2C\log g$
. Let
$|u|\leqslant q^{-\unicode[STIX]{x1D70E}}$
. Then
$$\begin{eqnarray}{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})\biggl(1+O\biggl(\frac{1}{Cg^{C-1}\log g}\biggr)\biggr).\end{eqnarray}$$
On the circle
$u=q^{-1/2}e(\unicode[STIX]{x1D703})$
, if
$\min _{1\leqslant j\leqslant 2g}\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert \geqslant c/2g$
with
$c>0$
and
$K\geqslant g^{2}/c$
, then
$$\begin{eqnarray}{\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})={\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})\biggl(1+O\biggl(\frac{g^{2}}{cK}\biggr)\biggr).\end{eqnarray}$$
Proof. By Theorem 2 and the definition of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
,
$$\begin{eqnarray}\displaystyle {\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D}) & = & \displaystyle \frac{1}{2}(P_{K}(u,\unicode[STIX]{x1D712}_{D})+(qu^{2})^{g}P_{K}(\overline{u},\unicode[STIX]{x1D712}_{D}))\biggl(1+O\biggl(\frac{1}{Cg^{C-1}\log g}\biggr)\biggr).\nonumber\end{eqnarray}$$
Equation (31) now follows from the definition of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
. The proof of (32) is the same except that one uses Theorem 3.◻
Corollary 14. For
$|u|\leqslant q^{-1/2}$
,
$u\neq u_{j},j=1,2,\ldots ,2g$
,
$$\begin{eqnarray}\lim _{K\rightarrow \infty }{\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})\rightarrow {\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D}).\end{eqnarray}$$
9 The zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
Since
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
is a good approximation of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
, one wonders whether their zeros are close to one another or whether there are other connections between them. We have seen that both functions satisfy the Riemann hypothesis, so that is a good start. As previously, we write
$u_{j}=q^{-1/2}e(\unicode[STIX]{x1D703}_{j}),j=1,2,\ldots ,2g$
, for the zeros of
${\mathcal{L}}(u,\unicode[STIX]{x1D712}_{D})$
, where
$0\leqslant \unicode[STIX]{x1D703}_{1}\leqslant \unicode[STIX]{x1D703}_{2}\leqslant \cdots \leqslant \unicode[STIX]{x1D703}_{2g}<1$
. We denote the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
by
$v_{j}=q^{-1/2}e(\unicode[STIX]{x1D719}_{j}),j=1,2,\ldots ,$
with
$0\leqslant \unicode[STIX]{x1D719}_{1}\leqslant \unicode[STIX]{x1D719}_{2}\leqslant \cdots <1$
, leaving open for now the question of their number.
By (30), a necessary and sufficient condition for
$v_{j}=q^{-1/2}e(\unicode[STIX]{x1D719}_{j})$
to be a zero of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
is that
$$\begin{eqnarray}1+(qv_{j}^{2})^{g}\frac{P_{K}(\overline{v}_{j},\unicode[STIX]{x1D712}_{D})}{P_{K}(v_{j},\unicode[STIX]{x1D712}_{D})}=0.\end{eqnarray}$$
This is equivalent to
$$\begin{eqnarray}\displaystyle \text{e}^{4\unicode[STIX]{x1D70B}\text{i}g\unicode[STIX]{x1D719}_{j}+2\text{i}\arg P_{K}(v_{j},\unicode[STIX]{x1D712}_{D})}=-1 & & \displaystyle \nonumber\end{eqnarray}$$
or
$$\begin{eqnarray}\displaystyle & \displaystyle 2g\unicode[STIX]{x1D719}_{j}+\frac{1}{\unicode[STIX]{x1D70B}}\arg P_{K}(v_{j},\unicode[STIX]{x1D712}_{D})=2g\unicode[STIX]{x1D719}_{j}+S_{K}(\unicode[STIX]{x1D719}_{j},\unicode[STIX]{x1D712}_{D})\equiv \frac{1}{2}\;(\text{mod}\;1). & \displaystyle \nonumber\end{eqnarray}$$
Now, as
$\unicode[STIX]{x1D719}$
varies from
$0$
to
$1$
, the graph of the continuous curve
$$\begin{eqnarray}f_{K}(\unicode[STIX]{x1D719})=2g\unicode[STIX]{x1D719}+S_{K}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D712}_{D})\end{eqnarray}$$
traverses a vertical distance greater than or equal to
$f_{K}(1)-f_{K}(0)=2g-0=2g$
. Thus, it intersects at least
$2g$
of the horizontal lines
$y=k+\frac{1}{2},k\in \mathbb{Z}$
, possibly more than once. We let these values be
$\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2},\ldots$
in increasing order. Then the points
$v_{j}=q^{-1/2}e(\unicode[STIX]{x1D719}_{j})$
are the distinct zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
. Thus,
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
has
$2g$
zeros, counting multiplicities, and
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
has at least
$2g$
distinct zeros. Similarly, we see that the number of zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
on any arc
$u=q^{-1/2}e(\unicode[STIX]{x1D719}),0\leqslant \unicode[STIX]{x1D719}\leqslant \unicode[STIX]{x1D703}$
, where
$0\leqslant \unicode[STIX]{x1D703}<1$
, is
$$\begin{eqnarray}\displaystyle N_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\geqslant 2g\unicode[STIX]{x1D703}+S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})+O(1). & & \displaystyle \nonumber\end{eqnarray}$$
Combining this with Theorem 10, we obtain the following result.
Theorem 15. Let
$K\geqslant g\log g/\unicode[STIX]{x1D6F7}(g)$
. Then
$$\begin{eqnarray}\displaystyle N_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\geqslant 2g\unicode[STIX]{x1D703}+O(\unicode[STIX]{x1D6F7}(g)). & & \displaystyle \nonumber\end{eqnarray}$$
Next we show that the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
are close to those of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
when
$K$
is large. We saw that
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
has a zero at
$u=q^{-1/2}e(\unicode[STIX]{x1D703})$
if and only if
$$\begin{eqnarray}f_{K}(\unicode[STIX]{x1D703})=2g\unicode[STIX]{x1D703}+S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})\equiv {\textstyle \frac{1}{2}}\;(\text{mod}\;1).\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle f_{K}(\unicode[STIX]{x1D703}) & = & \displaystyle 2g\unicode[STIX]{x1D703}+S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})+(S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D}))\nonumber\\ \displaystyle & = & \displaystyle N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})+(S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})).\nonumber\end{eqnarray}$$
Suppose now that
$\unicode[STIX]{x1D703}_{i}$
and
$\unicode[STIX]{x1D703}_{i+1}$
are arguments corresponding to distinct consecutive zeros of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
, and let
$0<\unicode[STIX]{x1D6E5}<\frac{1}{2}(\unicode[STIX]{x1D703}_{i+1}-\unicode[STIX]{x1D703}_{i})$
. Then, on the interval
$I=[\unicode[STIX]{x1D703}_{i}+\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D703}_{i+1}-\unicode[STIX]{x1D6E5}]$
,
$N(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})$
is an integer. Thus, if
$|S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})|<\frac{1}{2}$
on
$I$
, then (34) cannot hold. By (22), if
$\unicode[STIX]{x1D703}\in I$
, we have
$$\begin{eqnarray}\displaystyle |S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})| & {\leqslant} & \displaystyle \mathop{\sum }_{j=1}^{2g}\min \bigg(\frac{1}{2},\frac{1}{4\unicode[STIX]{x1D70B}K\Vert \unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j}\Vert }\bigg)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{g}{2\unicode[STIX]{x1D70B}K\unicode[STIX]{x1D6E5}}.\nonumber\end{eqnarray}$$
Therefore, if
$K>g/\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6E5}$
, then
$S_{K}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})-S(\unicode[STIX]{x1D703},\unicode[STIX]{x1D712}_{D})<\frac{1}{2}$
on
$I$
. We have thus proved the following result.
Theorem 16. Let
$\unicode[STIX]{x1D703}_{i}$
and
$\unicode[STIX]{x1D703}_{i+1}$
correspond to distinct consecutive zeros of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
, and let
$0<\unicode[STIX]{x1D6E5}<\frac{1}{2}(\unicode[STIX]{x1D703}_{i+1}-\unicode[STIX]{x1D703}_{i})$
. Then, if
$K>g/\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6E5}$
,
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
has no zero on the interval
$I=[\unicode[STIX]{x1D703}_{i}+\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D703}_{i+1}-\unicode[STIX]{x1D6E5}]$
. In particular, the zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
cluster around the zeros of
${\mathcal{F}}(u,\unicode[STIX]{x1D712}_{D})$
as
$K\rightarrow \infty$
.
Our last theorem concerns the simplicity of zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
. We may write
$$\begin{eqnarray}{\mathcal{F}}_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})={\textstyle \frac{1}{2}}P_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})(1+e(f_{K}(\unicode[STIX]{x1D703})))\end{eqnarray}$$
with
$f_{K}(\unicode[STIX]{x1D703})$
defined in (33). Recall that
$\unicode[STIX]{x1D703}$
corresponds to a zero of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
if and only if (34) holds. This zero will be simple if and only if
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{d}{d\unicode[STIX]{x1D703}}{\mathcal{F}}_{K}(q^{-1/2}e(\unicode[STIX]{x1D703}),\unicode[STIX]{x1D712}_{D})\neq 0, & \displaystyle \nonumber\end{eqnarray}$$
which is easily seen to be equivalent to
$$\begin{eqnarray}\displaystyle \frac{df_{K}(\unicode[STIX]{x1D703})}{d\unicode[STIX]{x1D703}}\neq 0. & & \displaystyle \nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \displaystyle \frac{df_{K}(\unicode[STIX]{x1D703})}{d\unicode[STIX]{x1D703}}=2g+2\mathop{\sum }_{j=1}^{2g}\bigg(\mathop{\sum }_{k=1}^{K}\cos (2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{j})k)\bigg). & & \displaystyle \nonumber\end{eqnarray}$$
The right-hand side is a trigonometric polynomial in
$\unicode[STIX]{x1D703}$
of degree
$K$
, so it has at most
$2K$
zeros. Now, by Theorem 15, if
$K\geqslant g\log g/\unicode[STIX]{x1D6F7}(g)$
, then
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
has
${\geqslant}2g(1+o(1))$
zeros. Thus, if we also have
$K=o(g)$
, then at most
$o(g)$
of these will be multiple. Taking
$\unicode[STIX]{x1D6F7}(g)=g/\text{log}_{q}g$
, we deduce the following result.
Theorem 17. If
$\log g\log _{q}g\leqslant K=o(g)$
, then almost all zeros of
${\mathcal{F}}_{K}(u,\unicode[STIX]{x1D712}_{D})$
are simple.
